1. Introduction

Graphene has attracted intense interest in photonics, electronics, and optoelectronics due to its unique electronic and optical properties [1, 2]. The linear optical conductivity takes the universal value [3] σ₀ = e²/4h̄ over a very broad energy range, due to the linear dispersion at low energy excitations. In the dipole approximation, second order nonlinearities are absent in pristine graphene due to the inversion symmetry of the honeycomb lattice, but third order nonlinearities are observed to be several orders of magnitude larger than in conventional semiconductors and metals [2, 4–10]. Various third-order processes have been measured, including four wave mixing [4], third harmonic generation [5, 6], self-phase modulation [2, 7, 8], two-photon absorption [9], and two-color coherent current injection [10]. The electrical properties of graphene are also striking, especially at low carrier densities. The mobility [11] can be as high as 2 × 10⁶cm²V⁻¹s⁻¹ for carrier densities below 5 × 10⁹cm⁻²; in that density regime a minimum conductivity [12–14] on the order of 4e²/πh has been observed experimentally. At higher carrier densities, the electron transport is diffusive, with the dc resistance [15] mainly due to electron scattering from local and remote impurities, and electron-phonon scattering; this is well described by Boltzmann transport theory [16, 17]. Experimentally, the density and hence the conductivity can be tuned by chemical doping [18] or by applying an external gate voltage [19].

In the development of integrated graphene-based photonic devices, the presence of a second order nonlinearity is important for realizing various essential building blocks, such as fast and compact electro-optic modulators [20, 21], optical switches [22–24], and optical transistors [25, 26]. In many of these device implementations the second-order process being exploited is second harmonic generation (SHG). For instance, several commonly proposed schemes to obtain all-optical switching and transistor action rely on employing a SHG-signal as seed or output signal [22,23,25,26]. Another example is the use of SHG for enhancing thermo-optical switching in ring resonators [24]. SHG in graphene-based photonic devices can arise from breaking the inversion symmetry of graphene, by growing graphene on a substrate to utilize the surface effect [27, 28], by utilizing the weak variation of the optical electric field over the unit cell and its associated magnetic field [29–33], or by applying a dc electric field to generate an asymmetric steady state [34, 35].

This work focuses on the last scenario, for which the effective second order susceptibility ${\chi }_{\text{eff}}^{(2);dab}({\omega }_1,{\omega }_2)$ can be described by a third order response ${\chi }^{(3);dabc}({\omega }_1,{\omega }_2,0)E_{\text{dc}}^c$ with $E_{\text{dc}}^c$ the dc field strength. In describing the dc limit ω₃ → 0 of ${\chi }^{(3);dabc}({\omega }_1,{\omega }_2,{\omega }_3)$ for any material, one of two following regimes is usually appropriate: (i) in an insulator or undoped semiconductor, a dc field induces a static electric polarization, which gives rise to a nonzero second order optical response; this is usually called electric field induced second harmonic generation (EFISH); (ii) in a doped semiconductor or graphene, besides inducing a static electric polarization the dc field drives free electrons to generate a dc current, which can result in current induced second harmonic generation (CSHG). When describing a steady dc current, the inclusion of current relaxation is essential to avoid divergences. Thus at small ω₃ the third order nonlinearity coefficient can be written as ${\chi }^{(3);dabc}({\omega }_1,{\omega }_2,{\omega }_3)={\chi }_J^{dabc}({\omega }_1,{\omega }_2,{\omega }_3)+{\chi }_E^{dabc}({\omega }_1,{\omega }_2,0)+O({\omega }_3)$. Here the first term, with χ_J ∝ (ω₃ + i/τ)⁻¹, describes the current induced portion (CSHG), where a Drude model is used for the dc current with a relaxation time τ, while χ_E describes the EFISH. In our previous work [36], an analytic expression for σ^(3);dabc(ω₁, ω₂, ω₃) in doped graphene was obtained neglecting all scattering processes, and it does show a divergence as ${\omega }_3^{-1}$ when ω₃ → 0.

CSHG was first theoretically discussed in a doped semiconductor by Khurgin [37]. Wu et al. [34] applied the same method to treat bilayer graphene, and found that the effective current induced SHG can reach ${\chi }_{\text{eff}}^{(2)}~{10}^{-7}\hspace{0.17em}\text{m}/\text{V}$ for a surface current density around 6.6 A/m. This nonlinearity coefficient is about 5 orders of magnitude larger than that of quartz [38], 2–3 orders of magnitude larger than calculated values for monolayer BN [39], and comparable to the value of the monolayer material MoS₂ [40]. In monolayer graphene the CSHG has already been experimentally observed [35], and intraband contributions to it have been theoretically studied [33]. However, an evaluation of the contribution from interband transitions, which are usually important in the optical frequency regime, is still lacking. A full theoretical treatment is of interest as monolayer graphene could be a very suitable material for generating strong CSHG, since the maximum surface current density in monolayer graphene can be up to 3000 A/m [41], which is several order of magnitude higher than that in regular semiconductors.

In this paper, we theoretically consider the dc current induced second order nonlinearity in doped monolayer graphene while including the interband transitions. We first introduce the semiconductor Bloch equations (SBE) to describe the dc and optical response of graphene under relaxation time approximations. We calculate the shift of the carrier distribution in the Brillouin zone that is induced by the dc field, and then turn to the second-order nonlinear optical response and give an analytic expression for the nonlinear susceptibility. From this result we estimate the CSHG coefficients at different frequencies and relaxation times, and compare the results with those of other materials.

2. Model

We start with the SBE for graphene, which we treat in the usual two-band tight binding model [42]. In the presence of a total electric field E_tot(t) = E_dc + E(t), the SBE are written as

Before the optical field is turned on, we assume the dc field drives the system to a steady state ${\rho }_{\mathit{k}}^{\text{dc}}$. Even here, in a full microscopic theory it would be important to take into account carrier-carrier scattering, carrier-phonon scattering, and carrier-impurity scattering, leading to a very complicated calculation [48,49]. But within the framework of Eq. (1), for a weak dc field the shifted populations n_sk+δk, with δk ≈ eE_dcτ₁/h̄ constitute a good approximation to steady state; they follow from a perturbative solution of the steady state equation

With the optical field applied, the system responds as ${\rho }_{\mathit{k}}(t)={\rho }_{\mathit{k}}^{\text{dc}}+{\rho }_{\mathit{k}}^{\text{op}}(t)$, with the contribution to the density matrix due to the optical excitation, ${\rho }_{\mathit{k}}^{\text{op}}(t)$, satisfying the equation

3. Results

We now use these equations to calculate the current induced second order nonlinearities in doped graphene with optical transitions around the Dirac cones K and K′. Each of these cones provides the same contribution, so we only consider the conductivity induced by transitions around the K point explicitly and double it to get the total result, including also a factor of 2 to for the spin degeneracy. As in our previous work [36], we use a tight binding model to describe the electron states in the π (s = −) and π^* (s = +) bands, and use the standard approximations around the Dirac points: eigenenergies are ε_sK+k ≈ sh̄v_Fk with Fermi velocity $v_F=\sqrt{3}{a}_0{\gamma }_0/2\overline{h}$ where γ₀ = 2.7 eV [42] is the nearest neighbour hopping energy; velocity matrix elements are v_ssK+k ≈ sv_Fk/k and v_ss̄K+k ≈ isv_Fẑ × k/k; the Berry connections are ξ_++k = ξ_−−k and ξ_ss̄K+k ≈ ẑ × k/2k². In the expressions for the response coefficients given below, and in the plots of them, we take the zero temperature limit.

To derive the correct linear response in the undoped limit, we reinstate $\delta {\rho }_{\mathit{k}}^N$ and, within our approximations, Eq. (3) and (4) are given by

Turning to the nonlinear response in doped graphene, with the neglect of EFISH ( $\delta {\rho }_{\mathit{k}}^N\to 0$) we find that the current induced second order conductivity can be written as ${\sigma }_{\text{eff};J}^{(2);dab}({\omega }_1,{\omega }_2)={\sigma }_J^{dabc}({\omega }_1,{\omega }_2)J_{\text{dc}}^c$ with ${\sigma }_J^{dabc}({\omega }_1,{\omega }_2)=-\frac{{(\overline{h}{v}_{F}e)}^2}{8\pi {\left|\mu \right|}^4}{S}^{dac}\left(\frac{\overline{h}{\omega }_1}{\left|\mu \right|},\frac{\overline{h}{\omega }_2}{\left|\mu \right|},\frac{\overline{h}{\tau }_2^{-1}}{\left|\mu \right|}\right)$. Here the dimensionless quantity S^dabc is a fourth order tensor. Considering that the point group of the graphene monolayer is D_6h (6/mmm), there are only three independent components S^xxyy, S^xyxy, and S^xyyx [51], which are given as

 

Fig. 1 w dependence of S^xxyy(w, w, γ) [(a) real part and (c) imaginary part] and S^xyyx(w, w, γ) [(b) real part and (d) imaginary part] for different γ. Insets in (c) and (d) illustrate the resonant optical transitions of the peak at w = 1 and w = 2 respectively. In the inset, the band structure is illustrated around the Dirac cones, the arrows are for photons with frequency given aside, the short green lines indicate the intermediate states, the orange dashed lines show the chemical potential without dc field, and the filled orange regions indicate the shifted distribution under dc field.

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An effective SHG coefficient can be constructed according to ${\chi }_{\text{eff}}^{(2);dab}={\sigma }_J^{(3);dabc}{J}_{\text{dc}}^c/(2\omega {\epsilon }_0{d}_{\text{gr}})$, with d_gr = 3.3 Å the effective thickness of graphene. By taking τ₁ = 1 ps and $E_{\text{dc}}^x=0.1\hspace{0.17em}\text{kV}/\text{cm}$ yielding a k-space shift δk = 1.5×10⁷/m, which is the same as in Wu et al.’s calculation [34], the estimated surface dc current density is J_dc = 1.1 × 10³ A/m, much larger than the value in bilayer graphene due to the difference between the parabolic and linear band structures. For a comparison, we take the chemical potential μ = 0.2 eV and the optical damping ${\mathrm{\Gamma }}_2=\overline{h}{\tau }_2^{-1}=0.05\hspace{0.17em}\text{eV}$ from Wu et al.; the peak value at w = 1, which corresponds to a fundamental wavelength of 6.2 μm, is ${\chi }_{\text{eff}}^{(2);xxx}~1.2\times {10}^{-7}\hspace{0.17em}\hspace{0.17em}\text{m}/\text{V}$ which is comparable to Wu et al.’s result for bilayer graphene. However, for a chemical potential μ = 0.53 eV, which corresponds to a shorter fundamental wavelength of 2.35 μm, we find a lower peak value of ${\chi }_{\text{eff}}^{(2);xxx}=2.5\times {10}^{-9}\hspace{0.17em}\hspace{0.17em}\text{m}/\text{V}$ for Γ₂ = 0.05 eV and ${\chi }_{\text{eff}}^{(2);xxx}=1.24\times {10}^{-8}\hspace{0.17em}\hspace{0.17em}\text{m}/\text{V}$ for Γ₂ = 0.01 eV. The peak CSHG coefficients at higher fundamental frequencies are smaller due to the |μ|⁻⁴ dependence of ${\sigma }_J^{dabc}$.

4. Discussion and conclusion

We have calculated the current induced second order response in graphene by using the semiconductor Bloch equations under relaxation time approximations. Our general equations hold for arbitrary temperatures, but we exhibited expressions for the response coefficients and plotted their values in the zero temperature limit. These depend strongly on the scattering time (both at dc and at optical frequencies), the chemical potential, and the fundamental frequency. Since the thermal energy at room temperature is about 30 meV, which is much smaller than the Fermi energies in which we are interested (about 0.5 eV), the effects of finite temperature can be expected to be negligible, except insofar as they affect the scattering times; taking the limit to zero temperature allowed us to give analytic expressions for the response coefficients, which should lead to simple estimates and an improved physical understanding. A maximum value of the σ^(2);dab is found when the energy of the fundamental photon matches the chemical potential. For a surface current density 10³ A/m in graphene, the effective ${\chi }_{\text{eff}}^{(2);xxx}$ can be as large as 10⁻⁷m/V for h̄ω = 0.2 eV or 10⁻⁸ m/V for h̄ω = 0.53 eV. For the same parameters, the coefficient of CSHG is comparable to that in bilayer graphene [34]. For the usual scenario of graphene on a substrate there is always the possibility of interface effects also leading to SHG, and there will be forbidden contributions due to deviations from the electric dipole approximation in the calculation of the optical response. But CSHG can be controlled by adjusting the dc current, and so it should be possible both to separate out any CSHG contribution from the total SHG, and as well control the total SHG, by adjusting the dc current [35]. Thus our calculations show that the use of dc current to break inversion symmetry and make graphene second-harmonic active should lead to the effective generation of second harmonic light in graphene structures.